Asymmetric Power Hazard Distribution for COVID-19 Mortality Rate under Adaptive Type-II Progressive Censoring: Theory and Inferences

This article investigates the estimation of the parameters for power hazard function distribution and some lifetime indices such as reliability function, hazard rate function, and coefficient of variation based on adaptive Type-II progressive censoring. From the perspective of frequentism, we derive the point estimations through the method of maximum likelihood estimation. Besides, delta method is implemented to construct the variances of the reliability characteristics. Markov chain Monte Carlo techniques are proposed to construct the Bayes estimates. To this end, the results of the Bayes estimates are obtained under squared error and linear exponential loss functions. Also, the corresponding credible intervals are constructed. A simulation study is utilized to assay the performance of the proposed methods. Finally, a real data set of COVID-19 mortality rate is analyzed to validate the introduced inference methods.


Introduction
To achieve a balance between the total time spent in the experiment and the number of units used in the experiment, the experiment must be subjected to a good control system (censoring scheme) that allows the experimenter to obtain results that enable him to make good statistical inference. At the same time, the working experimental units are saved for future use, as well as the cost and time associated with testing. Type-I (time) and Type-II (failure of units) censoring schemes are the oldest and most common censoring schemes. In these two types of censoring schemes, units cannot be withdrawn from an experiment until the final stage or the number of units fails. Balakrishnan and Sandhu [1] introduced the progressive Type-II censoring that has more flexibility than the Type-II censoring in allowing units to be withdrawn from the test at different observed failure times. Briefly, it can be described as follows: assume that n independent units are placed in the life test and the observed failure times m(m ≤ n), which are called a progressive sample as X 1:m:n < X 2:m:n < . . . < X m:m:n , are prefixed. Besides, the scheme R � (R 1 , R 2 , . . . , R m ) is a prefixed censoring plan. At the first failure occurrence X 1 , the surviving units R 1 are randomly withdrawn from the test; then, at the second failure occurrence X 2 , the surviving units R 2 are randomly withdrawn from the test and so on until the occurrence of the m th failure and therefore the remaining surviving units R m � n − m − m− 1 i�1 R i are withdrawn from the test and the test is terminated; see Figure 1. For more details and extensive reviews of the literature on progressive Type II censoring, readers may refer to Balakrishnan [2], Balakrishnan and Sandhu [3], and EL-Sagheer [4]. Kundu and Joarder [5] introduced a Type-II progressive hybrid censoring scheme; the experiment in this type is terminated at a prescribed time T with a progressive Type-II right censored. It is clear that this type of censoring has the same disadvantage of the Type-I censoring (time), in which m is random and may be a small number. For this reason, Ng et al. [6] developed the adaptive Type-II progressive censoring scheme as a mix of Type-II progressive censoring and Type-I censoring. is type allows R 1 , R 2 , . . . , R m to depend on the failure times so that the effective sample size is always m, which is fixed in advance. us, the life testing experiment can save both the total test time and the cost induced by failure of the units.
is censoring scheme can be described as follows: consider n identical units under observation in a life testing experiment and suppose the experimenter provides a time T, which is an ideal total test time, but we may allow the experiment to run over time T. If the m th progressively censored observed failure occurs before time T (i.e., X m: m: n < T), the experiment terminates at the time X m: m: n . Otherwise, once the experimental time passes time T but the number of observed failures has not reached m, we would want to terminate the experiment as soon as possible. erefore, we should leave as many surviving units on the test as possible. Suppose J is the number of failures observed before time T, that is, X J:m:n ≤ T < X J+1:m:n , J � 0, 1, . . . , m, where X 0:m:n ≡ 0 and X m+1:m:n ≡ ∞. After the experiment passed time T, we set is formulation leads us to terminate the experiment as soon as possible if the (J + 1)th failure time is greater than T for J + 1 < m. One extreme case is when T ⟶ ∞, which means time is not the main consideration for the experimenter; then we will have a usual progressive Type-II censoring scheme with the prefixed progressive censoring scheme (R 1 , . . . , R m ). Another extreme case can occur when T � 0, which means we always want to end the experiment as soon as possible; then, we will have R 1 , . . . , R m− 1 � 0 and R m � n − m, which results in the conventional Type-II censoring scheme. For extensive reviews of the literature on the adaptive Type-II progressive censoring scheme, readers may refer to Cramer and Iliopoulos [7], Mohie El-Din et al. [8], and El-Sagheer et al. [9], which can be briefly described by Figures 2 and 3.
From Figures 1 and 2, we observe that the value of T plays an important role in the determination of the values of R i , i � 1, 2, . . . , m, and is a compromise between a shorter experimental time and a higher chance to observe extreme failures.
Mugdadi [10] proposed the two-parameter power hazard function distribution, denoted by PHFD (θ, λ) as an alternative to the Weibull, Rayleigh, and exponential distributions and studied its different properties.
is distribution has increasing hazard rate function (HRF) for λ > 0, and it has decreasing HRF for − 1 < λ < 0. e probability density function (PDF), cumulative distribution function (CDF), reliability function RF, and HRF are given, respectively, by where θ and λ are the scale and shape parameters, respectively. e PHFD has been shown to be useful for modeling and analyzing the life time data in medical and biological sciences, engineering, and so on.
is distribution is a very flexible model that approaches different models when its parameters are changed. It contains the following special models: PHFD refers to Rayleigh (β) when θ � 1/β 2 and λ � 1, PHFD reduces to Weibull (θ, 1) when λ � θ − 1, and PHFD is exponential distribution with mean 1/θ when λ � 0. e coefficient of variation is used in many science areas such as medical sciences, chemistry, biology, economics, finance, engineering, and reliability theory to assess homogeneity of bone test samples, dose-response studies, wildlife studies, and other fields; see Subrahmanya Nairy and Rao [11], Creticos et al. [12], Versaci et al. [13], and El-  Sagheer et al. [9]. Given a set of observations from PHFD (θ, λ), the sample coefficient of variation (CV) is where E(X) and E(X 2 ) are the first and the second moments of the PHFD (θ, λ), given by where where Inferences for the PHFD have been studied by several authors including Kınacı [14] discussed the stress-strength reliability model for PHFD, El-Sagheer [15] investigated the problem of point and interval estimations of the parameters for PHFD based on progressive Type-II censoring scheme, Mugdadi and Min [16] discussed Bayes estimation of PHFD based on a complete and Type-II censored samples, and Khan [17] established the recurrence relations for single and product moments of generalized order statistics from the PHFD. Recently, El-Sagheer et al. [18] constructed the Bayesian and non-Bayesian approaches for the lifetime performance index with the progressive Type-II censored sample from the PHFD. In this article, we use the adaptive Type-II progressive censored (AT2PC) scheme to study the problem of estimating the shape and scale parameters, RF, HRF, andCV for two-parameter PHFD. e structure of the paper is organized as follows. Section 2 deals with the maximum likelihood estimate and asymptotic confidence intervals. Bayesian estimates using Markov chain Monte Carlo technique are provided in Section 3. In Section 4, a simulation study is conducted to compare the performance of these estimation methods. A real data set of COVID-19 is presented to illustrate the application of the proposed inferences in Section 5. Finally, a brief conclusion is given in Section 6.

Maximum Likelihood Inference
Suppose that x � X R 1: m: n < X R 2: m: n < . . . < X R m: m: n is a AT2PC-order statistics from the PHFD with progressive censored scheme R � (R 1 , R 2 , . . . , R m ). According to Ng et al. [6], the log-likelihood function ℓ � log L(θ, λ; x) without normalized constant can be written as where x i is used instead of x R i: m: n: k . After differentiating ℓ with respect to θ and λ, respectively, and equating each of them to zero, the likelihood equations can be written as follows: From (10), the MLE of θ can be illustrated as Since (10) and (12) do not have closed-form solutions, Newton-Raphson iteration method is widely used to obtain the desired MLEs in such situations. For more details, see El-Sagheer [4]. Once MLEs of θ and λ are obtained, the MLEs of r(t), h(t), and CV for a given mission time t can be obtained after replacing θ and λ by θ and λ according to the invariant property of the MLEs as 2.1. Approximate Confidence Intervals. According to Cohen [19], the Fisher information matrix (FIM), which is defined by the negative expectation of the partial second derivative of the log-likelihood function, is needed to construct the approximate confidence intervals (ACIs) for parameters Λ � (θ, λ).
Computational Intelligence and Neuroscience 3 e elements of the FIM are Since MLE has asymptotic normality property under certain regularity conditions, the estimator Λ � (θ, λ) has asymptotic distribution Λ − Λ ⟶ N(0, I − 1 (Λ)), and the inverse matrix of I(Λ) is us, the 100(1 − c)% ACIs of Λ is where is the percentile of the standard normal distribution with right-tail probability c/2.

Delta Method.
In order to construct the ACIs of r(t), h(t), and CV, we need to find the variances of them. So, Greene [20] implemented the delta method for this purpose. In this method, for analytically computing the variance, a linear approximation of the functions of the MLEs is created, and then the variance of the simpler linear function that can be used for large sample inference is computed. Let W � (zΔ/zθ, zΔ/zλ) be the first partial derivative of Δ � (r(t), h(t), CV). en, the approximate estimates of variances Δ � (r(t), h(t), CV) can be written as where W T is the transpose matrix of W. us, the (1 − c)100% ACIs for Δ can be given by Moreover, Meeker and Escobar [21] suggested using the natural approximation for the log-transformed MLE to prevent the negative lower bound of the ACIs. us, twosided (1 − c)100% normal ACIs for Θ � θ, λ, r(t), h(t) or CV are given by 4 Computational Intelligence and Neuroscience

Bayesian Inference
To characterize the problems more rationally and reasonably, we must take into consideration both the information from observed sample data and the prior information; this is the main idea of Bayesian inference. In this section, Bayesian inference procedures using Markov chain Monte Carlo (MCMC) technique are proposed to estimate the parameters θ and λ as well as r(t), h(t), and CV under both squared error (SE) and linear exponential (LINEX) loss functions. Also, the corresponding CRIs are constructed under MCMC technique. We consider here that the parameters θ and λ follow the gamma prior distributions with PDFs where a 1 , b 1 , a 2 and b 2 reflect the knowledge of prior about (θ, λ) and they are assumed to be known and nonnegative hyperparameters. In addition, the parameters θ and λ as well as the corresponding priors are considered here to be independent. Further, the joint prior of the parameters θ and λ can be written as Via Bayes' theorem across using the likelihood function L(θ, λ; x) with the joint prior π(θ, λ), the joint posterior density π * (θ, λ| x) of θ and λ can be obtained as Under SE loss function, the Bayesian estimator of any function of θ and λ, say φ(θ, λ), is given by Under LINEX loss function, we have where c ≠ 0. It is clear that the ratio of two integrals in (25) and (26) cannot be obtained in a closed form. us, it is necessary to use suitable numerical methods to approximate these integrals. So, in this case, we apply MCMC technique to obtain the Bayes estimates of θ, λ, r(t), h(t), and CV and corresponding credible intervals (CRIs).

Markov Chain Monte Carlo
Technique. MCMC technique is one of the most general technique for an estimation, which is provided here to compute the Bayes estimates and the corresponding CRIs for θ, λ, r(t), h(t), and CV. It is known that there are several procedures of MCMC technique available in which samples are generated from the conditional posterior densities. One of the simplest MCMC procedures is the Gibbs sampling procedure, which was proposed by Geman and Geman [22]. Another procedure is considered the Metropolis-Hastings (M-H) algorithm, which was proposed by Metropolis et al. [23] and later extended by Hastings [24]. A more general procedure of MCMC procedures which we will use is considered the M-H within Gibbs sampling. Gibbs sampler is required to decompose the joint posterior distribution into full conditional distributions for each parameter and then sample from them. From (26), the posterior conditional density function of θ given λ can be written as Similarly, Computational Intelligence and Neuroscience Moreover, the conditional posterior density of θ given in (27) is gamma density with shape parameter (m + a 1 ) and scale parameter us, by implementing any gamma generating routine, samples of θ can be simply generated. In addition, the conditional posterior density of λ cannot be reduced analytically to well-known distribution. So, according to Tierney [25], M-H algorithm within Gibbs sampling with normal proposal distribution is used to conduct the MCMC methodology. e hybrid M-H algorithm and Gibbs sampler works as follows: (1) Start with an (θ (0) � θ, λ (0) � λ), and set k � 1.

Simulation Study
Monte Carlo simulations were implemented utilizing 1000 AT2PC samples for each simulation to compare the estimators of parameters θ and λ as well as some lifetime parameters r(t), h(t), and CV discussed in the preceding sections. By using Mathematica ver. 12, all computations were conducted. We conducted a simulation study using different combinations of T, n, and m and different censored scheme R. We used the algorithm proposed by Ng et al. [6] to simulate an AT2PC sample from the PHFD with parameters (θ, λ) � (2, 1.5). e true values of r(t), h(t), and CV at time t � 0.4 are evaluated to be 0.9222, 0.506, and 0.4279, respectively. e performance of estimators is evaluated in terms of mean square error (MSE), which is computed as  2000) as "burnin." In addition, we assume the informative gamma priors for θ and λ, that is, when the hyperparameters are a i � 1 and b i � 2, i � 1, 2. Moreover, 95% CRIs were computed for each simulated sample. In our study, we consider two different values of T � 3, 5 and the following censoring schemes (CS):

Application of COVID-19 Data
In this section, for illustrative purposes, a real-life data for the coronavirus is presented to inspect the inference procedures discussed in the previous sections. We consider the set of real-life data, which is reported by Almongy et al. [26].
is data represents a COVID-19 mortality rate data that belongs to Italy of 59 days, which is recorded from 27 February to 27 April 2020. e data are as follows: (33) For the goodness of fit test, we compute the Kolmogorov-Smirnov (K-S) distances between the empirical distribution and the fitted distribution functions. e K-S is 0.09254 and the associated p-value is 0.73124. erefore, according to the p-value, we can say that the PHFD fits quite well to the above data. Empirically, Q − Q and P − P plots are shown in Figure 4, which clearly show that the PHFD fits the data very well.  Under the previous data, we use the algorithm proposed by Ng et al. [6] to generate an AT2PC sample with m � 30, T � 5, and R � {3, 0, 2, 0, 1, 0, 2, 0, 0, 3, 0, 3, 0, 1, 2, 0, 0, 3, 0, 0, 2, 1, 0, 3, 0, 1, 0, 2, 0, 0}. us, the resulting AT2PC sample is as follows: Based on the previous sample of AT2PC, the MLEs and ACIs for θ, λ, r(0.4), h(0.4), and CV are determined to be as in Table 6. Moreover, to compute the Bayesian estimates, the prior distributions of the parameters need to be specified.   Empirical and theoretical dens.

Computational Intelligence and Neuroscience
Since we have no prior information, we assume that noninformative gamma priors for θ and λ, that is, when the hyperparameters are a i � 0.0001 and b i � 0.0001, i � 1, 2. Under MCMC technique, the posterior analysis was done across combining M-H algorithm within Gibbs sampler. To conduct the MCMC algorithm which was described in Section 3.1, the initial values of the parameters θ and λ were taken to be their MLEs. In addition, 12000 MCMC samples were generated. To avoid the effect of the initial values (starting point), we expunge the first 2000 samples as "burnin." Table 6 shows the Bayesian estimates as well as 95% CRIs for θ, λ, r(0.4), h(0.4), and CV.

Conclusion
e purpose of this paper is to develop different methods to estimate the unknown quantities θ, λ, r(t), h(t), and CV of the PHFD using AT2PC scheme, which is introduced by Ng et al. [6]. e MLEs as well as ACIs using asymptotic distributions are obtained. Furthermore, to obtain the CIs of the reliability characteristics and coefficient of variation, we used delta method. It is clear that, after studying the Bayesian estimates, the posterior distribution equations of the unknown quantities are complicated and so hard to reduce analytically to well-known forms. For this reason, we have applied MCMC technique to compute the Bayes estimators. e Bayes estimates have been computed under both SE and LINEX loss functions. For illustrative purpose, real data set of COVID-19 mortality rates is considered. To check and compare the performance of the proposed methods, a simulation study was implemented with different sample sizes (T, n, m) and different CSs (I, II, III). According to the results, we note the following: (1) As expected, from Tables 1-5, as sample sizes (n, m) increases, the MSEs and ALs decrease.  Data Availability e datasets generated during the current study are available from the corresponding author on reasonable request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.